A uniform thin bar of mass \(6~\mathrm{kg}\) and length \(2.4\) meter is bent to make an equilateral hexagon. The moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of hexagon is:
1. \(0.8\mathrm{~kg} \mathrm{~m}^2\)
2. \(0.4\mathrm{~kg} \mathrm{~m}^2\)
3. \(0.1\mathrm{~kg} \mathrm{~m}^2\)
4. \(8\mathrm{~kg} \mathrm{~m}^2\)

Four point masses, each of mass m are fixed at the corners of a square of side \(l\). The square is rotating with angular frequency \(\omega\), about an axis passing through one of the corners of the square and parallel to its diagonal, as shown in the figure. The angular momentum of the square about this axis is:

 
1. \( 2 m l^2 \omega \)
2. \( m l^2 \omega \)
3. \( 3 m l^2 \omega \)
4. \( 4 m l^2 \omega \)

Shown in the figure is a hollow ice cream cone (it is open at the top). If its mass is \(M\), radius of its top, \(R\) and height, \(H\), then its moment of Inertia about its axis is:

   
1. \( \frac{M\left(R^2+H^2\right)}{4} \)
2. \( \frac{M R^2}{3} \)
3. \( \frac{M R^2}{2} \)
4. \( \frac{M H^2}{3} \)

Two solid uniform circular discs of the same material and equal thickness have radii \(R_1=R\) and \(R_2=\alpha R.\) The moments of inertia of the discs about their respective central axes perpendicular to their planes are \(I_1\)​ and \(I_2​.\) If \(I_1 : I_2 = 1 : 16,\) then the value of \(\alpha\) is:
1. \(\sqrt{2} \)
2. \(4\)
3. \(2\)
4. \(2\sqrt{2} \)

For a uniform rectangular sheet shown in the figure, if \(I_O\) and \(I_{O'}\) be moments of inertia about the axes perpendicular to the sheet and passing through \(O\) (the centre of mass) and \(O'\) (corner point), then:

A particle of mass \(m\) is moving along side of a square of side '\(a\)', with a uniform speed \(v\) in the x-y plane as shown in the figure:

              
Which of the following statements is false for the angular momentum \(\vec L\) about the origin?

A cord is wound around the circumference of the wheel of radius \(r.\) The axis of the wheel is horizontal and the moment of inertia about it is \(I.\) A weight \(mg\) is attached to the cord at the end. The weight falls from rest. After falling through a distance \('h',\) the square of the angular velocity of the wheel will be:
1. \( \dfrac{2 m g h}{I+2 m r^2} \)

2. \( \dfrac{2 m g h}{I+m r^2} \)

3. \( 2 g h\)

4. \( \dfrac{2 g h}{I+m r^2} \)

Two masses \(A\) and \(B,\) each of mass \(M\) are fixed together by a massless spring. A force acts on the mass \(B\) as shown in the figure. If mass \(A\) starts moving away from mass \(B\) with acceleration \(a,\) then the acceleration of mass \(B\) will be:

Distance of the centre of mass of a solid uniform cone from its vertex is \(Z_0\). If the radius of its base is \(R\) and its height is \(h\) then \(Z_0\) is equal to:
1. \( \frac{{h}^2}{4{R}} \)
2. \(\frac{3 h}{4} \)
3. \(\frac{5 h}{8} \)
4. \(\frac{3{h}^2}{8{R}}\)

From a solid sphere of mass \(M\) and radius \(R\) a cube of maximum possible volume is cut. The moment of inertia of a cube about an axis passing through its center and perpendicular to one of its faces is:
1. \( \frac{{MR}^2}{32 \sqrt{2 \pi}} \)
2. \( \frac{{MR}^2}{16 \sqrt{2} \pi} \)
3. \( \frac{4 {MR}^2}{9 \sqrt{3} \pi} \)
4. \( \frac{4{MR}^2}{3 \sqrt{3} \pi}\)